In the signum function, the value is (-1) for negative (x). In exams, keep the three cases (x<0), (x=0), and (x>0) separate.
Step 2
Why this answer is correct
The correct answer is A. ( -1 ). In the signum function, the value is (-1) for negative (x). In exams, keep the three cases (x<0), (x=0), and (x>0) separate.
Step 3
Exam Tip
साइनम फलन में ऋणात्मक (x) के लिए मान (-1) होता है। परीक्षा में (x<0), (x=0), (x>0) तीन स्थितियां अलग रखें।
Since (f(-x)=f(x)), the graph is symmetric about the (y)-axis. In exams, check (f(-x)) to identify an even function.
Step 2
Why this answer is correct
The correct answer is B. (y)-अक्ष / (y)-axis. Since (f(-x)=f(x)), the graph is symmetric about the (y)-axis. In exams, check (f(-x)) to identify an even function.
Step 3
Exam Tip
(f(-x)=f(x)) इसलिए ग्राफ (y)-अक्ष के प्रति सममित है। परीक्षा में सम फलन पहचानने के लिए (f(-x)) जांचें।
\(\lfloor 2.7\rfloor=2\) because it is the greatest integer less than or equal to (2.7). In exams, do not round; move down to the greatest integer.
Step 2
Why this answer is correct
The correct answer is A. ( 2 ). \(\lfloor 2.7\rfloor=2\) because it is the greatest integer less than or equal to (2.7). In exams, do not round; move down to the greatest integer.
Step 3
Exam Tip
\(\lfloor 2.7\rfloor=2\) क्योंकि यह (2.7) से छोटा या बराबर सबसे बड़ा पूर्णांक है। परीक्षा में दशमलव को निकटतम पूर्णांक नहीं, नीचे वाले पूर्णांक में बदलें।
\(x^3\) is an odd function because (f(-x)=-f(x)). In exams, origin symmetry usually indicates an odd function.
Step 2
Why this answer is correct
The correct answer is B. क्योंकि (f(-x)=-f(x)) / because (f(-x)=-f(x)). \(x^3\) is an odd function because (f(-x)=-f(x)). In exams, origin symmetry usually indicates an odd function.
Step 3
Exam Tip
\(x^3\) विषम फलन है क्योंकि (f(-x)=-f(x))। परीक्षा में मूल बिंदु सममिति के लिए विषम फलन देखें।
A. (x)-अक्ष के समानांतर रेखा/line parallel to (x)-axis
Step 1
Concept
In (f(x)=3), (y=3) for every (x). In exams, treat a constant function as a horizontal line.
Step 2
Why this answer is correct
The correct answer is A. (x)-अक्ष के समानांतर रेखा / line parallel to (x)-axis. In (f(x)=3), (y=3) for every (x). In exams, treat a constant function as a horizontal line.
Step 3
Exam Tip
(f(x)=3) में हर (x) के लिए (y=3) रहता है। परीक्षा में स्थिर फलन को क्षैतिज रेखा मानें।
((x-3)) means shifting the graph (3) units to the right. In exams, read (x-a) as a right shift.
Step 2
Why this answer is correct
The correct answer is B. (3) इकाई दाईं ओर / (3) units right. ((x-3)) means shifting the graph (3) units to the right. In exams, read (x-a) as a right shift.
Step 3
Exam Tip
((x-3)) का अर्थ ग्राफ को दाईं ओर (3) इकाई खिसकाना है। परीक्षा में (x-a) को दाईं ओर विस्थापन समझें।
For \(\sqrt{x-4}\), the smallest (x) is (4) and then (y=0). In exams, start a square-root graph by making the inside expression (0).
Step 2
Why this answer is correct
The correct answer is B. ( (4,0) ). For \(\sqrt{x-4}\), the smallest (x) is (4) and then (y=0). In exams, start a square-root graph by making the inside expression (0).
Step 3
Exam Tip
\(\sqrt{x-4}\) में सबसे छोटा (x=4) है और तब (y=0) है। परीक्षा में वर्गमूल ग्राफ की शुरुआत अंदर की राशि (0) से करें।
The horizontal asymptote of \(\frac{1}{x}\) is (y=0), and (+1) shifts it to (y=1). In exams, apply vertical shifts to asymptotes too.
Step 2
Why this answer is correct
The correct answer is B. (y=1). The horizontal asymptote of \(\frac{1}{x}\) is (y=0), and (+1) shifts it to (y=1). In exams, apply vertical shifts to asymptotes too.
Step 3
Exam Tip
\(\frac{1}{x}\) का क्षैतिज आसमापी (y=0) है और (+1) से वह (y=1) हो जाता है। परीक्षा में ऊर्ध्व विस्थापन आसमापी पर भी लागू करें।
In \(y=\frac{1}{x}\), (x) and (y) have the same sign. In exams, remember that the reciprocal graph lies in quadrants (1) and (3).
Step 2
Why this answer is correct
The correct answer is B. \(y=\frac{1}{x}\). In \(y=\frac{1}{x}\), (x) and (y) have the same sign. In exams, remember that the reciprocal graph lies in quadrants (1) and (3).
Step 3
Exam Tip
\(y=\frac{1}{x}\) में (x) और (y) का चिह्न समान होता है। परीक्षा में पारस्परिक फलन की शाखाएं चतुर्थांश (1) और (3) में याद रखें।
For \(y=x^2\), \(y\ge 0\) and the graph lies on both sides of the (y)-axis. In exams, identify the parabola by its (U)-shape.
Step 2
Why this answer is correct
The correct answer is A. \(y=x^2\). For \(y=x^2\), \(y\ge 0\) and the graph lies on both sides of the (y)-axis. In exams, identify the parabola by its (U)-shape.
Step 3
Exam Tip
\(y=x^2\) में \(y\ge 0\) होता है और ग्राफ (y)-अक्ष के दोनों ओर होता है। परीक्षा में परवलय को (U)-आकार से पहचानें।
C. पूरे ( \(-\infty,\infty\) ) पर/on all ( \(-\infty,\infty\) )
Step 1
Concept
The graph of \(x^3\) rises continuously from left to right. In exams, remember \(x^3\) is increasing on the whole real domain.
Step 2
Why this answer is correct
The correct answer is C. पूरे ( \(-\infty,\infty\) ) पर / on all ( \(-\infty,\infty\) ). The graph of \(x^3\) rises continuously from left to right. In exams, remember \(x^3\) is increasing on the whole real domain.
Step 3
Exam Tip
\(x^3\) का ग्राफ बाएं से दाएं लगातार ऊपर जाता है। परीक्षा में \(x^3\) को पूरे वास्तविक प्रांत में बढ़ता हुआ याद रखें।
The graph of (|x|) decreases up to (x=0) and then increases. In exams, see the left arm of the (V)-graph as decreasing.
Step 2
Why this answer is correct
The correct answer is A. ( \(-\infty,0] \). The graph of (|x|) decreases up to (x=0) and then increases. In exams, see the left arm of the (V)-graph as decreasing.
Step 3
Exam Tip
(|x|) का ग्राफ (x=0) तक नीचे आता है और फिर ऊपर जाता है। परीक्षा में (V)-ग्राफ का बायां भाग घटता हुआ देखें।
A. ( (-2,0) ) और ( (2,0) )/( (-2,0) ) and ( (2,0) )
Step 1
Concept
On the (x)-axis, set (y=0), giving \(x^2-4=0\). In exams, set the related coordinate to (0) for intercepts.
Step 2
Why this answer is correct
The correct answer is A. ( (-2,0) ) और ( (2,0) ) / ( (-2,0) ) and ( (2,0) ). On the (x)-axis, set (y=0), giving \(x^2-4=0\). In exams, set the related coordinate to (0) for intercepts.
Step 3
Exam Tip
(x)-अक्ष पर (y=0) रखकर \(x^2-4=0\) मिलता है। परीक्षा में अवरोधों के लिए संबंधित निर्देशांक (0) रखें।
\(x^2+1\) is always at least (1), so (y=0) is impossible. In exams, use the minimum value to check (x)-intercepts.
Step 2
Why this answer is correct
The correct answer is A. (0) बार / (0) times. \(x^2+1\) is always at least (1), so (y=0) is impossible. In exams, use the minimum value to check (x)-intercepts.
Step 3
Exam Tip
\(x^2+1\) हमेशा (1) या उससे बड़ा है इसलिए (y=0) नहीं हो सकता। परीक्षा में न्यूनतम मान से (x)-अवरोध जांचें।
The slope of (2x) is (2), greater than the slope (1) of (x). In exams, treat the coefficient of (x) as the slope in a linear function.
Step 2
Why this answer is correct
The correct answer is B. अधिक ढाल वाली रेखा / line with greater slope. The slope of (2x) is (2), greater than the slope (1) of (x). In exams, treat the coefficient of (x) as the slope in a linear function.
Step 3
Exam Tip
(2x) की ढाल (2) है जो (x) की ढाल (1) से अधिक है। परीक्षा में रैखिक फलन में (x) के गुणांक को ढाल मानें।
For \(y=\sqrt{x}\), \(x\ge 0\) and \(y\ge 0\). In exams, associate the square-root graph with the first quadrant.
Step 2
Why this answer is correct
The correct answer is A. \(y=\sqrt{x}\). For \(y=\sqrt{x}\), \(x\ge 0\) and \(y\ge 0\). In exams, associate the square-root graph with the first quadrant.
Step 3
Exam Tip
\(y=\sqrt{x}\) में \(x\ge 0\) और \(y\ge 0\) होता है। परीक्षा में वर्गमूल ग्राफ को पहले चतुर्थांश से जोड़ें।
For \(x^2=\sqrt{x}\), (x=0) and (x=1) work. In exams, test simple possible points first.
Step 2
Why this answer is correct
The correct answer is C. ( (0,0) ) और ( (1,1) ) / ( (0,0) ) and ( (1,1) ). For \(x^2=\sqrt{x}\), (x=0) and (x=1) work. In exams, test simple possible points first.
Step 3
Exam Tip
समानता \(x^2=\sqrt{x}\) के लिए (x=0) और (x=1) सही हैं। परीक्षा में संभावित सरल बिंदुओं को पहले जांचें।
\(\frac{1}{x^2}\) is always positive and \(x\ne 0\). In exams, when \(x^2\) is in the denominator, remember (y) is positive.
Step 2
Why this answer is correct
The correct answer is A. पहले और दूसरे / first and second. \(\frac{1}{x^2}\) is always positive and \(x\ne 0\). In exams, when \(x^2\) is in the denominator, remember (y) is positive.
Step 3
Exam Tip
\(\frac{1}{x^2}\) हमेशा धनात्मक होता है और \(x\ne 0\) होता है। परीक्षा में \(x^2\) हर में हो तो (y) धनात्मक याद रखें।
\(y=\frac{1}{4}\) is obtained by substituting (x=4). In exams, put (x) in the denominator for a reciprocal function.
Step 2
Why this answer is correct
The correct answer is B. \(\frac{1}{4}\). \(y=\frac{1}{4}\) is obtained by substituting (x=4). In exams, put (x) in the denominator for a reciprocal function.
Step 3
Exam Tip
\(y=\frac{1}{4}\) सीधे (x=4) रखने से मिलता है। परीक्षा में पारस्परिक फलन में (x) को हर में रखें।
The vertex of (y=|x|) is ((0,0)), and (-3) shifts it downward. In exams, add vertical shifts directly to the (y)-coordinate.
Step 2
Why this answer is correct
The correct answer is C. ( (0,-3) ). The vertex of (y=|x|) is ((0,0)), and (-3) shifts it downward. In exams, add vertical shifts directly to the (y)-coordinate.
Step 3
Exam Tip
(y=|x|) का शीर्ष ((0,0)) है और (-3) से नीचे खिसकता है। परीक्षा में ऊर्ध्व विस्थापन सीधे (y)-निर्देशांक में जोड़ें।
The graph \(y=\frac{1}{x}\) approaches the axes as asymptotes. In exams, remember both asymptotes of the reciprocal graph.
Step 2
Why this answer is correct
The correct answer is C. \(y=\frac{1}{x}\). The graph \(y=\frac{1}{x}\) approaches the axes as asymptotes. In exams, remember both asymptotes of the reciprocal graph.
Step 3
Exam Tip
\(y=\frac{1}{x}\) अक्षों के पास जाता है लेकिन उन्हें आसमापी की तरह नहीं काटता। परीक्षा में पारस्परिक ग्राफ के दोनों आसमापी याद रखें।
For vertical stretching, multiply the whole function by (2). In exams, treat an outside multiplier as vertical scaling.
Step 2
Why this answer is correct
The correct answer is B. \(y=2x^2\). For vertical stretching, multiply the whole function by (2). In exams, treat an outside multiplier as vertical scaling.
Step 3
Exam Tip
ऊर्ध्व खिंचाव में पूरे फलन को (2) से गुणा करते हैं। परीक्षा में बाहरी गुणक को ऊर्ध्व खिंचाव मानें।
Because the coefficient is negative, the parabola opens downward and the vertex (y)-value (4) is maximum. In exams, the vertex is maximum for a downward parabola.
Step 2
Why this answer is correct
The correct answer is C. (4). Because the coefficient is negative, the parabola opens downward and the vertex (y)-value (4) is maximum. In exams, the vertex is maximum for a downward parabola.
Step 3
Exam Tip
ऋणात्मक गुणांक के कारण परवलय नीचे खुलता है और शीर्ष का (y)-मान (4) अधिकतम है। परीक्षा में नीचे खुलने वाले परवलय में शीर्ष अधिकतम होता है।
\(y=x^3\) is negative for negative (x) and positive for positive (x). In exams, remember the cubic graph crosses the (x)-axis.
Step 2
Why this answer is correct
The correct answer is C. \(y=x^3\). \(y=x^3\) is negative for negative (x) and positive for positive (x). In exams, remember the cubic graph crosses the (x)-axis.
Step 3
Exam Tip
\(y=x^3\) ऋणात्मक (x) पर ऋणात्मक और धनात्मक (x) पर धनात्मक होता है। परीक्षा में घन ग्राफ को (x)-अक्ष पार करते हुए याद रखें।
For the (x)-axis, (|x+1|-2=0) gives (|x+1|=2). In exams, open the modulus into both cases.
Step 2
Why this answer is correct
The correct answer is B. (x=-3) और (x=1) / (x=-3) and (x=1). For the (x)-axis, (|x+1|-2=0) gives (|x+1|=2). In exams, open the modulus into both cases.
Step 3
Exam Tip
(x)-अक्ष के लिए (|x+1|-2=0) से (|x+1|=2) मिलता है। परीक्षा में मापांक को खोलकर दोनों स्थितियां हल करें।
For \(\sqrt{x+5}\), \(x+5\ge 0\), so \(x\ge -5\). In exams, keep the expression inside the square root at least (0).
Step 2
Why this answer is correct
The correct answer is A. \( [-5,\infty\) ). For \(\sqrt{x+5}\), \(x+5\ge 0\), so \(x\ge -5\). In exams, keep the expression inside the square root at least (0).
Step 3
Exam Tip
\(\sqrt{x+5}\) के लिए \(x+5\ge 0\) इसलिए \(x\ge -5\)। परीक्षा में वर्गमूल के अंदर की राशि को (0) या उससे बड़ा रखें।
In a constant function, (y) is always (2). In exams, the range of a constant line is the set containing only that (y)-value.
Step 2
Why this answer is correct
The correct answer is A. ( {2} ). In a constant function, (y) is always (2). In exams, the range of a constant line is the set containing only that (y)-value.
Step 3
Exam Tip
स्थिर फलन में (y) हमेशा (2) रहता है। परीक्षा में स्थिर रेखा का परिसर केवल उसी (y)-मान का सेट होता है।
B. यह (y)-अक्ष के प्रति सममित है/it is symmetric about the (y)-axis
Step 1
Concept
The domain of \(y=x^2\) is all real numbers and its graph is symmetric about the (y)-axis. In exams, remember the domain and range of a parabola separately.
Step 2
Why this answer is correct
The correct answer is B. यह (y)-अक्ष के प्रति सममित है / it is symmetric about the (y)-axis. The domain of \(y=x^2\) is all real numbers and its graph is symmetric about the (y)-axis. In exams, remember the domain and range of a parabola separately.
Step 3
Exam Tip
\(y=x^2\) का प्रांत सभी वास्तविक संख्याएं है और ग्राफ (y)-अक्ष के प्रति सममित है। परीक्षा में परवलय के प्रांत और परिसर अलग-अलग याद रखें।
